So, I can make a brief comment or two. However, I'll note that this particular sub field is a little outside my specialty. I can point to a few resources however.
First, a small reading list that may or may not be helpful:
J. J. Collins, Beyond the Qumran Community: The Sectarian Movement of the Dead Sea Scrolls (Grand Rapids: Wm. B. Eerdmans, 2009).
J. J. Collins, Between Athens and Jerusalem: Jewish Identity in the Hellenistic Diaspora (second edition; Eerdmans, 2000).
L. Grabbe, A History of the Jews and Judaism in the Second Temple Period, 2 vols. (T&T Clark, 2004).
Part of the problem with much of this time period (at least the Persian pd., ca. 540-330 BCE or so) is that we have very little data for most things. I suppose one of the biggest resources for understanding what conversion may have looked like would be the Dead Sea Scrolls, where we have some textual data for at least one sect of Judaism during the time frame you're asking about. The Rule of the Community (1QS) details the kinds of things new memebers of the community were expected to do, delineating various time periods to wait and participate in things like communal living, eating, possessions, etc.
Part of the issue is there has never been any such thing as a singular, normative form of Judaism. There have always been sects and "denominations" and what not. The majority of which, and someone with better data than me feel free to correct me, probably did not allow for conversion. This was mostly an identity issue (identity being one of the primary motivating factors for various aspects of Second Temple period social norms). So, eating pig during the Hellenistic period became a major identity thing, as did circumcision (this worked both ways, and people sometimes tried to "uncircumcise" themselves...I'll leave that to your imagination). Take, for example, 1 Macc 1:14-15:
So they built a gymnasium in Jerusalem, according to Gentile custom, and removed the marks of circumcision, and abandoned the holy covenant. They joined with the Gentiles and sold themselves to do evil.
I'm sure there's stuff all over Josephus and other comparable texts (including the New Testament). But that's about what I can offer for now.